Semiconductor substrates such as silicon and III-V materials (cg: GaAs) are the foundation for building semiconductor devices. This discussion will refer to silicon for its popularity and anisotropic properties; however, the reference to the material ‘silicon’ can be replaced with any material type (isotropic or anisotropic). Semiconductor substrates such as silicon are crystal lattice structures. Silicon conducts electricity in a very controlled manner relative to the impurities put into its crystal lattice structure making silicon an excellent surface to build electronic devices on. Semiconductor wafers are cut from ingots. Ingots are made from pure polysilicon (polycrystalline silicon) chips (processed from sand) and melted at high temperatures (eg, 1400c). A single crystal (seed) is lowered into molten silicon contained in a pure argon gas. A small amount of silicon rises with the rotating silicon seed and cools into a perfect monocrystalline ingot (Single Crystal Silicon Ingot). A silicon crystal is anisotropic, meaning its elastic (stiffness) properties are directionally dependent. Semiconductor substrates, called wafers, are cut from the ingot, machined and polished. An orientation mark (flat or notch) is made on the wafer edge indicating the orientation the wafer is cut relative to the crystal (crystallographic) orientation in the ingot. The orientation is noted as the substrate type (eg: Si<100> or Si<111>) described by Miller Indices. The anisotropicity of the crystal causes non-spherical deformation of semiconductor substrates (eg: silicon wafers) from stress, regardless of the orientation the wafer is sliced from the ingot however the stiffness (stress constant) is different for different wafer types (eg: Si<100>=180500, Si<111>=229200) and required in the stress equation.
The traditional (practiced) method of determining the stress in a thin film is by measuring the change in the curvature of a substrate the film is deposited onto. The radii of the surface before film deposition (R1) and after film deposition (R2) can be measured with an apparatus such as ‘Cheng 201’ (Cheng, 1993, U.S. Pat. No. 5,233,201). The radius of curvature can be calculated from 2 surface points using equation (1) R=2b(Δx/Δz):
(1) R=2b(Δx/Δz)R=2bΔx/Δz  (1)                eg: (2×300 mm×80 mm)/2.4 mm=20000 mm or 20 meters        
The unstressed and stressed radius values (R1, R2) are used to calculate film stress (σ) Young's modulus (or tensile modulus, Es) and Poisson's Ratio (v), named after Simeon Poisson, associate a stress constant (C=Es/(1−v)) to a substrate material type. The stress constant is used in the Stoney equation. Equation (2) Stoney Curvature is for a spherical shape with the same curvature across any section of the surface (any 2 points).
                              (          2          )                ⁢                                  ⁢        Stoney        ⁢                                  ⁢        Curvature                                                                      σ          =                                    (                                                ct                  s                  2                                                  6                  ⁢                                                                          ⁢                                      t                    f                    2                                                              )                        ×                          (                                                1                                      R                    2                                                  -                                  1                                      R                    1                                                              )                                      ⁢                                  ⁢                                                                              eg                  ⁢                                      :                                    ⁢                                                                          ⁢                  σ                                =                                ⁢                                                      (                                                                  180500                        ×                                                  0.55                          2                                                                                            6                        ×                                                  6000                          angstroms                                                ×                        0.0000001                                                              )                                    ×                                      (                                                                  1                        20000                                            -                                              1                                                  -                          200000                                                                                      )                                                                                                                          =                                ⁢                                  834                  ⁢                                                                          ⁢                  Mpa                                                                                        (        2        )            
Calculating stress of a thin film deposited onto an anisotropic substrate (eg: silicon) from curvature, conflicts with the definition of anisotropic materials. A discrepancy in stress measurement occurs from the non-spherical deformation of anisotropic polycrystalline substrates such as silicon noted in ‘G. C. A. M. Janssen, et al., Thin Solid Films (2008)’. The deformation of anisotropic substrates is described as the shape of a cylinder on its side, a saddle, potato chip and other (non spherical) shapes.
Building circuits on semiconductor substrates includes lithography, film deposition, etching, etc. to build patterns of circuitry (circuit architecture) that build complete circuits. Film thickness ranges from a few hundred to thousands of angstroms (eg: 500a to 15000a). The atomic structure of the substrate and the deposited films are different and cause stress from the different thermal expansion rates and elevated temperature of the film deposition process. Specifically, one material shrinks more than the other after film deposition causing stress. Stress causes the films to delaminate (peel, crack) and circuits in the devices (called ‘Die’) to fail. The lower yield raises the cost per device making it important to monitor thin film stress.
The discrepancy (inaccuracy) in ‘film stress measurements from curvature’ on anisotropic substrates can be observed using a film stress measurement tool (that calculates stress from curvature) and performing pre and post measurements at different surface locations (different angles related to the notch orientation or different distances from the substrate center). Both stress measurement methods (stress calculated from curvature and volume) are performed in FIG. 1 and FIG. 2. The ‘stress calculated from curvature’ changes when the curvature measurement points change. The ‘stress calculated from volume’ does not change as the substrate orientation changes. Another approach to correct the (curvature) discrepancy of anisotropic deformation is to mathematically combine radii samples from different areas of the substrate. The problem arises in (2) Stoney Curvature when the radius (R1 or R2) result=0. Specifically, if R=0 then 1/0=NaN (Not a Number), generating an error. Generally, selecting different points to calculate curvature (by rotating the wafer or positioning the points closer or further from the substrate center) derive different radii values that resolve different ‘stress calculated from radius’ values. The resettled volume difference technique (VDT) uses volume that remains continuously accurate at any orientation or starting point.
A new unstressed (no film deposited) ‘prime’ substrate is relatively flat (‘prime’ refers to the substrate specifications). Silicon is most spherical when it is unstressed, as if part of a large sphere. Anisotropic materials deform into less spherical shapes as stress increases. Generally, an anisotropic (silicon) substrate deforms from a (relatively) flat unstressed shape into a saddle shape as stress increases, making the results, from curvature, less accurate as stress increases. An actual anisotropic silicon substrate is shown at different angles in FIG. 1 and FIG. 2, illustrating the inaccuracy of (2) Stoney Curvature method as the stress measurement changes from −672 Mpa at 261 degrees to −493 Mpa at 171 degrees, (a difference of 179 Mpa) while the ‘stress calculated from volume’ is an unchanging −575 Mpa at any orientation or angle.
Other parameters such as bow height (BH) and warp are used to describe specifications of a substrate. The same inaccuracy in BH occurs when deriving BH from substrate curvature or radius from the following equation:
(3) BH from RBH=R±√{square root over (R2−r2)}  (3)
R=substrate surface ‘radius of curvature’, r=substrate diameter/2
Calculating the BH from volume is a more accurate description of the substrate using:
(4) BH from VBH=(2V/r2π)  (4)
Simply stated, Bow Height (Bow or Warp) is used to describe a stressed substrate. BH from curvature can be changed by relocating (rotating) the substrate in order to find a more or less flatter surface whereas using BH from volume resolves stable, unchanging values (that resolve radius, bow height and stress) in any orientation, providing an accurate substrate measurement and description. Substrate BH from volume, is calculated with equation (4) BH from V using volume within the geometric container FIG. 4, 4c (unstressed volume) or 4e (stresses volume).
Viewing semiconductor substrates in the 3D IME (FIG. 1) expedites a clear unambiguous understanding of anisotropic deformation, volume difference technique and how non-spherical deformation resolves inaccurate stress measurements from surface curvature. Surface shape of semiconductor substrates is not obvious due to the magnitude of height deformation compared to the length and width. The substrate appears flat in 3 equally magnified dimensions whatever the deformed shape as illustrated in FIG. 3. Generally, Using curvature to measure thin film stress on silicon is based on the belief that deformation is spherical (or closely spherical) because the user cannot see the deformed (non spherical) shape.